Multicomponent optical waveguide having index gradient

ABSTRACT

An optical waveguide for a communication system includes a graded index core formed from at least three glass-forming compounds with a profile having at least two α-type index profile terms. 
     The core has a refractive index which is n c  at the center of the core and which varies as a function of the radial distance r from the center of the core subststantially as: ##EQU1## where α i  is defined by: ##EQU2## where N≧2, 
     Δ=(n c   2  -n 0   2 )/2n c   2 , 
     n o  is the refractive index of said compounds at r=a, 
     N c  =n c  -λdn c  /dλ where λ is the wavelength of the light source, and the quantities Δ i  are parameters which can be varied provided the condition ##EQU3##  is satisfied.

BACKGROUND OF THE INVENTION

This invention relates to multimode optical waveguides having an optimumindex gradient and to methods of making them.

The propagation of light waves in optical waveguides is governed by lawsof physics similar to those that govern microwave propagation andtherefore can be studied in terms of modes, each of which has its ownpropagation and electromagnetic field characteristics. Single modewaveguides are advantageous in that they are capable of propagatingoptical signals with very low dispersion, but due to the low numericalaperture and/or small core size of such fibers, it is difficult toefficiently inject optical signals into these waveguides. Multimodewaveguides have larger core diameters and/or larger numerical aperturesthan single mode waveguides. Multimode waveguides are therefore oftenthe preferred medium for the transmission of optical signals since theycan efficiently accept light from injection lasers and incoherent, broadspectral width sources such as light emitting diodes. However, thousandsof modes propagate in multimode optical waveguides, each mode travelingat a slightly different group velocity. A short input pulse that isshared by many guided modes thus splits up into a sequence of pulsesthat arrive at the output end of the waveguide at different times. Thistype of pulse dispersion is the dominant cause of disperson in typicalmultimode optical waveguides.

Optical waveguides initially consisted of a core of uniform refractiveindex surrounded by a layer of cladding material having a lowerrefractive index. In this type of prior art fiber, the time required forthe various modes to travel a given longitudinal distance along thewaveguide increases as the mode order increases. The delay distortion insuch a fiber, defined as the difference in the times it takes thefastest mode and the slowest mode to traverse a given longitudinallength, is very large. Optical waveguides having cores with radiallygraded index profiles exhibit significantly reduced pulse dispersionresulting from group velocity differences among modes. This dispersionreducing effect, which is discussed in the publication by D. Gloge etal, entitled "Multimode Theory of Graded-Core Fibers," published in theNovember 1973 issue of the Bell System Technical Journal, pp. 1563-1578,employs a radially graded, continuous index profile from a maximum valueat the center of the core to a lower value at the core-claddinginterface. The index distribution in this type of waveguide is given bythe equation

    n(r)=n.sub.c [1-2Δ(r/a).sup.α ].sup.1/2 for r≦a

where n_(c) is the refractive index at the center of the core, n₀ is therefractive index of the fiber core at radius a, Δ=(n_(c) ² =n₀ ²)/2n_(c)² and a is the core radius.

It was initially thought that the parabolic profile wherein α is equalto 2 would provide an index gradient that would minimize dispersioncaused by group velocity differences among the modes.

The aforementioned Gloge et al publication describes a furtherdevelopment wherein a reduction in pulse dispersion is said to beobtained if, instead of α being equal to 2, it is equal to 2-2Δ.However, the theory concerning index gradients wherein α is equal to 2or 2-2Δ neglects effects introduced by differences between thedispersive properties of the core and cladding compositions.

U.S. Pat. No. 3,904,268--Keck and Olshansky describes a gradient indexoptical waveguide wherein the dispersive properties of the core andcladding are reduced. This patent teaches that the gradient indexoptical waveguide with minimal delay differences among the modes has anindex profile given by

    n.sup.2 (r)=n.sub.c.sup.2 [1-2Δ(r/a).sup.α ] r≦a

where ##EQU4## n_(c) is the refractive index at the center of the core,n₀ is the refractive index of the core at r=a, Δ=(n_(c) ² -n₀ ²)/2n_(c)² and N_(c) =n_(c) -λ₀ dn_(c) /dλ₀.

The invention of U.S. Pat. No. 3,904,268 is valid regardless of theglass composition provided the core refractive index is well describedby the foregoing over the spectral range over which the source operates.The technique of the patent is applicable for all binary ormulticomponent glass-forming compounds.

In accordance with the present invention, an additional class of gradedindex optical waveguides is described which are superior to the opticalwaveguides of U.S. Pat. No. 3,904,268 in their information carryingcapacity.

The wavelength dependence of pulse dispersion of optical waveguides isan important consideration. A waveguide which provides low pulsedispersion at several different wavelengths or over a range ofwavelengths is superior to one which provides low dispersion at or neara single wavelength. In the invention of U.S. Pat. No. 3,904,268, ingeneral, the waveguide has minimal dispersion at or near a singlewavelength. By choosing the profile shape of the waveguide according toKeck-Olshansky, minimal dispersion can be obtained at any chosenwavelength. However, as shown in FIG. (4) of this application, at otherwavelengths, the dispersion is significantly greater.

The article "Profile Synthesis in Multicomponent Glass Optical Fibers"by Kaminow and Presby, Applied Optics 16 January 1977 and U.S. Pat. No.4,025,156 of Gloge Kaminow and Presby show that by proper choice ofglass composition, an optical waveguide can be synthesized withdispersion minimized either over a range of wavelengths or at two ormore distinct distinct wavelengths.

U.S. Pat. No. 4,033,667, Fleming is related to the teachings of Kaminow,Presby and Gloge in teaching how a particular composition can have auniform index profile over a range of wavelengths.

As is clear from the examples cited in the Kaminow-Presby article, theGloge, Kaminow and-Presby patent, and the Fleming patent, theirinventions apply to only certain limited compositions. FIG. 1 in theKaminow-Presby paper shows that the P₂ O₅ concentration at r=o must be11.5 times greater than the GeO₂ concentration at r=a in order toachieve reduced pulse dispersion over an extended range of wavelengths.Although favorable from the viewpoint of dispersion, this composition isundersirable from the viewpoint of viscosity, thermal expansion,chemical durability and numerical aperture.

The same restrictions on composition are imposed by the teachings ofGloge and Presby. In their example, they find that the concentration ofGeO₂ at r=o must be eight times less than the concentration of B₂ O₃ atr=a. This restriction on composition makes it impossible to design anoptical fiber with other important properties such as high numericalaperture, good thermal expansion and viscosity matches across thediameter of the fiber.

The present invention avoids the severe restrictions on compositionwhich is required to practice the Gloge-Presby patent. As will be shown,a preferred embodiment of this invention is a graded index opticalwaveguide, having low dispersion over a range of wavelengths or at twoor more different wavelengths, and fabricated from a broad range ofpossible compositions.

As an example of the usefulness of the present invention, consider thefact that installing communication cables is very expensive. The cost ofoptical waveguides is quite small compared to this installation cost.The installed cables may have state of the art waveguides which haveminimum pulse dispersion at the wavelength of sources which arepresently being used, typically about 0.85 μ. In the future, sources maybe developed which are more efficient at other wavelengths. It would bevery desirable to use waveguides in cables presently being installedwhich will be capable of operating for a range of wavelengths. In thismanner, the cost of future installation of cables with waveguidescapable of operating at a different wavelength could be avoided.

SUMMARY OF THE INVENTION

In accordance with this invention, an optical waveguide is fabricatedfrom a plurality of glass-forming compounds with an index profile havingat least two index profile terms. The concentrations of these compoundsare varied so that the index of refraction is n_(c) at the center of thecore and varies as a function of the radial distance from the center ofthe core substantially as: ##EQU5## where α_(i) are index profilesdefined by: ##EQU6## where

N is greater than or equal to 2,

n_(c) is the refractive index at the center of the core,

n₀ is the refractive index of said compounds at the cladding r=a,

N_(c) =n_(c) -λdn_(c) /dλ, and

Δ=(n_(c) ² -n₀ ²)/2n_(c) ²

The quantities Δ_(i) are parameters which can be varied according toother requirements provided the condition ##EQU7## is satisfied.

According to another aspect of this invention, the quantities Δ_(i) canbe chosen so that minimized pulse dispersion is obtained over a range ofwavelengths or at several different wavelengths.

The foregoing and other objects, features and advantages of theinvention will be better understood from the following more detaileddescription and appended claims.

DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a segment of the waveguide of this invention;

FIG. 2 depicts its use in an optical communications system;

FIG. 3 depicts a method of fabricating the waveguide;

FIG. 4 shows the curve C1 depicting pulse dispersion versus wavelengthfor a conventional refractive index profile and the curve C2 depictingthe improved pulse dispersion versus wavelength for a preferredembodiment of this invention;

FIG. 5 shows the curve A0 which shows how the optimal α changes as afunction of λ for the single α profile; curves A1 and A2 show α₁ (λ) andα₂ (λ) for a preferred embodiment of the invention; both A1 and A2 havezero slope at 0.85 μm;

FIG. 6 shows curves B1 and G1 which show the B₂ O₃ and GeO₂concentration profiles for the single α profile designed for minimumdispersion at 0.85 μm; curves B2 and G2 show B₂ O₃ and GeO₂concentrations profile for a preferred embodiment of the double αprofile designed for dα_(i) /dλ=0 at 0.85 μm; and

FIG. 7 shows the curve C3 depicting pulse dispersion versus wavelengthfor a preferred embodiment of this invention; Low pulse dispersion isachieved at two separate wavelengths.

DESCRIPTION OF THE PREFERRED EMBODIMENT

A waveguide 11 has a core 12 and a cladding 13. The index of refractionof the cladding 13 is less than that of the core 12. The core 12 has agradient index of refraction which varies from n_(c) at the center ofthe core to n₀ at the radius a. In accordance with the present inventionthe gradient is formed from at least two index profile terms whichminimize pulse dispersion over a range of wavelength or at selectedwavelengths.

This waveguide is coupled into an optical communication system depictedin FIG. 2 wherein a transmitter 14 includes a source producing pulses oflight having a mean wavelength λ. A receiver 15 at the output end of thewaveguide receives light from the waveguide 11 and responds to thislight. It is desirable to minimize pulse dispersion between thetransmitter 14 and the receiver 14. Further, it is desirable to minimizedispersion over a range of wavelengths or at different wavelengths.

In accordance with this invention, the core 11 is made frommulticomponent glass, such as germania silicate and borosilicate glass.Each of these binary compounds has a concentration which varies radiallyin accordance with a different concentration profile.

Using the example of these compounds, n_(c) denotes the refractive indexof the germania silicate glass at r=0 and n₀ denotes the refractiveindex of the borosilicate glass at r=a.

In accordance with this invention, pulse dispersion in waveguidesfabricated from such multicomponent glass is minimized if the refractiveindex of the core varies as a function of the radial distance from thecenter of the core in accordance with: ##EQU8## where α_(i) are indexprofiles defined by: ##EQU9##

Values of n_(c), dn_(c) /dλ, n_(o), Δ_(i) and dn_(o) /dλ, which areneeded to produce a waveguide having such index profiles can be obtainedby performing a Sellmeier fit to refractive index measurements taken atdifferent wavelengths for glass compositions used as core and claddingmaterials.

Some techniques for measuring refractive index profiles of opticalwaveguides are disclosed in the publications: "Viewing Refractive-IndexProfiles and Small-Scale Inhomogeneities in Glass Optical Fibers: SomeTechniques" by C. A. Burrus et al., Applied Optics, Oct. 1974, Vol. 13,No. 10, pp. 2365-2369 and "Refractive Index Profile Measurements ofDiffused Optical Waveguides" by W. E. Martin, Applied Optics, September,1974, Vol. 13. No. 9, pp. 2112-2116 and in paper No. 3.5 entitled"Interferometric Measurement of SELFOC Dielectric Constant Coefficientsto Sixth Order" by E. G. Rawson et al., 1973, IEEE/OSA Conference onLaser Engineering and Applications, May 30-June 1, 1973, a briefdescription of which may be found on pp. 22-23 of the Digest ofTechnical Papers presented at this conference.

Theory

The theoretical analysis showing that pulse dispersion is minimized isbased on the analysis of Marcatili in Bell Systems Technical Journal 56,49 (1977). Consider the class of index profiles ##EQU10## where N is apositive integer. Marcatili shows that the delay time of mode μ,ν isgiven by

    τ.sub.μν =T(1-B.sub.μν /D)/(1-B.sub.μν).sup.1/2(5)

where

    T=LN.sub.c /C                                              (6)

    N.sub.c =n.sub.c -λdn.sub.c /dλ              (7) ##EQU11## where k=2π/λ, L is the fiber length, and C is the speed of light. Since B.sub.μν is a small quantity of order ##EQU12## to order Δ.sup.2, ##EQU13## If D=2-6/5Δ, the root mean square of the delay differences among the modes is minimized for N=1 and is equally reduced for N≧2.

Eq. (8) can be written as a partial differential equation, ##EQU14## Forthe class of profiles described in U.S. Pat. No. 3,904,268, N=1, andthis equation has the solution ##EQU15##

For profiles given by the more general expression, Eqs. (3)-(4), I havefound that new solutions to Eq. (11) exist if ##EQU16## Thusmulticomponent glass optical fibers can have their pulse dispersionminimized also by the profile of Eqs. (1)-(2).

In the foregoing analysis, note that the term ##EQU17## given in theKeck and Olshansky patent has been simplified to -₅ ¹² Δ. That is, ithas been assumed that |y|<1 which is a valid assumption for most cases.

The desirable condition of minimum dispersion over a range ofwavelengths can be obtained for the optimum index profile of thisinvention. This imposes the condition that: ##EQU18##

If the α_(i) have zero derivative, the minimum pulse dispersion isobtained over a broader band of wavelengths.

An alternative condition, which is desirable for certain applications,is that minimum dispersion be obtained at two (or more) differentwavelengths.

    α.sub.i (λ.sub.1)=α.sub.i (λ.sub.2) i=1 . . . N. (15)

EXAMPLE 1

As a first example consider a multimode optical waveguide consisting ofa fused silica core doped at the center with 7.9 mole % GeO₂ and dopedat r=a with 13.5 mole % B₂ O₃. Measurements of the refractive indices ofthe germania-silica glasses can be found in the paper by S. Kobayashi,S. Shibata, N. Shibata, T. Izawa appearing in the digest of the 1977International Conference on Integrated Optics and Optical FiberCommunications held in Tokyo, Japan. The refractive indices of theborosilica glass can be found in the paper by J. W. Fleming appearing inthe Journal of the American Ceramic Society 59, 503-507 (1976).

The refractive index data referred to above was measured on bulk samplesof glass. The refractive index of an optical waveguide fiber can besubstantially different from the refractive index of a bulk samplebecause of well known quenching effects which occur during the fiberdraw. All refractive indices referred to in this application relate tothe actual refractive index of an optical fiber. The refractive indexdata based on bulk sample measurements is used solely for the purpose ofillustrating the practice of this invention.

From the appropriate Sellmeier fits it can be found that for awavelength λ of 0.85 μm, n_(c) is equal to 1.46493, n_(o) is equal to1.45071, and Δ is equal to 0.00966.

Using the prior art single profile of U.S. Pat. No. 3,904,268 tominimize pulse dispersion at 0.85 μm, the α-value is equal 1.957. Theroot mean square pulse broadening in nanoseconds/kilometer (ns/km) forthis waveguide is shown by C1 as a function of λ in FIG. 4. A minimumpulse dispersion of 0.013 ns/km is achieved at 0.85 μm.

In one possible embodiment of this invention, Δ₁ and Δ₂ are chosen suchthat

    Δ.sub.1 =(n.sub.c.sup.2 -n.sub.s.sup.2)/2n.sub.c.sup.2

    Δ.sub.2 =(n.sub.s.sup.2 -n.sub.o.sup.2)/2n.sub.c.sup.2

where n_(s) is the refractive index of fused silica. n_(s) can becalculated from the Sellmeier fit reported by I. H. Malitson in theJournal of the Optical Society of America, 55, 1205 (1965). At 0.85 μm,n_(s) is equal to 1.45250. For this example Δ₁ is equal to 0.00845 andΔ₂ is equal to 0.00121. Again using Sellmeier fits we obtain the values:##EQU19## Using these values in equation (1), we obtain values of β₁=2.076 and α₂ =1.121 which approximately minimizes pulse dispersion at0.85 μm.

After the optimal values of α_(i) are determined, the refractive indexgradient of the core is specified by equation (1). An optical waveguidesatisfying equation (1) can then be formed in accordance with one of themethods disclosed in U.S. Pat. Nos. 3,823,995 Carpenter and 3,826,560Schultz, for example.

In general, where the refractive index varies linearly with dopantconcentration, the concentration profiles C_(j) (r) of p glass formingcompounds vary substantially as: ##EQU20## where the coefficients C_(ij)and the α_(j) are chosen to give reduced pulse dispersion according tothe consideration discussed herein, and where the concentrations areexpressed as either mole fractions, weight fractions, atomic fraction orany other measure of concentration.

Regardless of the method of fabrication, it is improbable that a gradedindex optical waveguide, wherein α is precisely equal to a predeterminedvalue, can be formed. However, it is possible to fabricate waveguideswherein the index profile approximates the optimal profile defined byequations (1) and (2) and yet achieve a significant reduction in pulsedispersion. These techniques are adapted to the fabrication ofmulticomponent glass waveguides in the manner depicted in FIG. 3 whichwill subsequently be described.

In the following examples 2 and 3, it is shown how preferred embodimentsof the invention, represented by either Eq. (14) or Eq. (15) can beimplemented. For simplicity it is assumed in these examples that thenumber of terms in Eq. (1) is equal to 2.

In these examples, we define two quantities which respectively representthe change in refractive index of fused silica caused by theintroduction of germania and the change caused by the introduction ofboron oxide. These two quantities are:

    δ.sub.G =n.sub.c.sup.2 -n.sub.s.sup.2                (16a)

    δ.sub.B =n.sub.s.sup.2 -n.sub.o.sup.2                (16b)

where n_(s) is the refractive index of fused silica. These twoquantities δ_(G) and δ_(B) are related to quantities corresponding withΔ₁ and Δ₂ in the preceding example. We define two further terms:##EQU21## The refractive index of fused silica, n_(s), can be calculatedfrom the Sellmeier fit determined by I. H. Malitson, J. Opt. Soc. Amer.55, 1205 (1965). In this example n_(c) is again taken to be therefractive index of silica doped with 7.9 mole percent GeO₂, and n_(o)the refractive index of silica doped with 13.5 mole percent B₂ O₃.Define the further quantities: ##EQU22## To divide the effect of the twodopants between Δ₁ and Δ₂, one can define Δ₁ and Δ₂ by

    Δ.sub.1 =[δ.sub.G (1-X.sub.G)+δ.sub.B X.sub.B ]/(2n.sub.c.sup.2)                                        (21a)

    Δ.sub.2 =[δ.sub.G X.sub.G +(1-X.sub.B)δ.sub.B ]/(2n.sub.c.sup.2)                                        (21b)

X_(G) is a measure of the amount of germania which is assigned to Δ₂. IfX_(G) is 1, all of the germania is assigned to Δ₂. X_(B) is a measure ofthe amount of boron oxide assigned to Δ₁. Equations (21a-b) areconsistent with the required condition

    Δ=(Δ.sub.1 +Δ.sub.2).                    (22)

In equations (21a-b), the quantities X_(G) and X_(B) are introduced tospecify Δ₁ and Δ₂. In many glass systems the square of the refractiveindex is proportional to dopant concentration. If this is the case thenX_(G) and X_(B) are proportional to the dopant concentrations of GeO₂and B₂ O₃. This proportionality, however, is not necessary for thepractice of this invention.

By using equations (16)-(21) it will now be shown that one can findX_(G) and X_(B) such that either of the preferred embodimentsrepresented respectively by equation (14) or equation (15) is specified.

EXAMPLE 2

Consider an example in which the index profiles are given by equations(1-2), the Δ_(i) 's are given by equations (16) and (21), and thedesirable condition producing minimum dispersion over a range ofwavelengths, equation (14), is met. This is equivalent to the condition:

    (1-X.sub.G).sup.2 A.sub.GG +(1-X.sub.G)X.sub.B A.sub.BG +X.sub.B.sup.2 A.sub.BB =0                                               (23a)

    (1-X.sub.B).sup.2 A.sub.BB +(1-X.sub.B)X.sub.G A.sub.BG +X.sub.G.sup.2 A.sub.GG =0                                               (23b)

In the foregoing, the A's are coefficients determined by the refractiveindices of the glasses with which we are working. Equations (23a) and(23b) can be expressed by the more general form:

    (1-X.sub.i).sup.2 A.sub.ii +(1-X.sub.i)X.sub.j A.sub.ij +X.sub.j.sup.2 A.sub.jj =0                                               (24)

i=G, j=B or i=B, j=G

where

    A.sub.ij =δ.sub.i 'δ.sub.j '-(δ.sub.i δ.sub.j "+δ.sub.j δ.sub.i ")/2-2Vδ.sub.i δ.sub.j V(δ.sub.i 'δ.sub.j +δ.sub.j δ.sub.i ')/2. (25)

In writing Eq. (24), small correction terms of order X³ have beenneglected.

The quantities A_(GG), A_(BG), A_(BB) of eq. (25) can be evalulatedusing the aforementioned Sellmeier fits to refractive index data.

Equations (24) thus represent a pair of simultaneous quadratic equationswhich determine the design parameters X_(G) and X_(B). The solutions ofeq. (24) are

    X.sub.G =±[2A.sub.BB +A.sub.BG ±D]/(2D)              (26a)

    X.sub.B =±[2A.sub.GG +A.sub.BG ±D]/(2D)              (26b)

where

    D=(A.sup.2 BG-4 A.sub.BB A.sub.GG).sup.1/2.                (27)

Using the aforementioned Sellmeier fits and choosing λ=0.85 microns, onecan use these equations to find the solution:

    X.sub.G =0.772 and X.sub.B =1.082.                         (28)

Equations (21) and (22) can then be used to find

    Δ.sub.1 =9.04×10.sup.-3 and Δ.sub.2 =0.62×10.sup.-3 (29a) ##EQU23## Equation (2) then gives the desired result:

    α.sub.1 =1.810 and α.sub.2 =4.088.             (30)

The index profile characterized by Equations (29) and (30) can beproduced if the germania and boron oxide dopant concentration profilesare:

    C.sub.G (r)=0.079[1.-0.772(r/a).sup.1.810 -0.228(r/a).sup.4.088 ](31a)

and

    C.sub.B (r)=0.135 [2.082(r/a).sup.1.810 -1.082(r/a).sup.4.088 ]. (31b)

These results are based on the assumption that the refractive indexvaries linearly with dopant concentration. For glass-forming systemsexhibiting departures from linearity, this invention can still beapplied by extending the analysis to include the non-linear terms.

FIG. 3 depicts apparatus for forming a waveguide having the profiles ofequations (30) and (31). The apparatus will be described first so thatthe manner in which the present invention is used to operate thisapparatus can be better understood.

A layer 16 of glass soot is applied to a substantially cylindrical glassstarting member or bait rod 17 by means of outside vapor phase oxidationburner 18. Fuel gas and oxygen or air are supplied to burner 18 from asource not shown by a suitable means such as pipe 19. This mixture isburned to produce flame 20 which is emitted from the burner.

Containers 21, 22 and 23 hold quantities of liquid constituents 24, 25and 26 respectively which will ultimately form layer 16. A suitablegaseous medium, such as oxygen or the like, is supplied to thecontainers and bubbled through the liquids by means of tubes 27, 28 and29. The gaseous medium or carrier gas is supplied from a suitablesource, not shown, in predetermined quantities and at predeterminedpressures. The flow of carrier gas which is bubbled through liquidconstituent 24 in container 21 is regulated by valve 30, the flow rateof this carrier gas being indicated by gauge 31. Similarly, the flows ofcarrier gas bubbled through liquid constituents 25 and 26 in containers22 and 23 are regulated by valves 32 and 33 with the flow rates of thesegases being indicated by gauges 34 and 35.

The liquid constituents in the containers are maintained at the desiredtemperatures by heaters. As the carrier gas is bubbled through theheated liquid constituents, vapors of this liquid become entrained inthe carrier gas and are exhausted by means of tube or pipe 36. Thecarrier gas vapor mixture is fed to outside vapor phase oxidation burner18 and is injected into flame 20 wherein the gas vapor mixture isoxidized to form a glass soot. The soot leaves flame 20 in a streamwhich is directed toward starting member 17. Starting member 17 is bothrotated and translated as indicated by the arrows adjacent thesupporting end of the starting member so that a uniform deposition ofsoot is applied to the starting member.

Containers 21-23 contain a glass forming reactants and at least twodopants. In this example, container 21 contains SiCl₄, container 25contains GeCl₄ and container 26 contains BCl₃.

The valves 30-32 are controlled in the manner described in the SchultzU.S. Pat. No. 3,826,560 to produce the gradient index of refraction.Valves 32 and 33 are controlled in accordance with this invention tovary the dopant concentration in the desired manner.

In accordance with the invention, control circuits 37 and 38 control theconcentrations of the two dopants while the waveguide preform is beingformed. A sensor 39 produces an electrical output representing theradius of the waveguide preform as it is being formed. This signal isapplied to each of the control circuits 37 and 38. Control circuit 37produces a control signal which varies in accordance with equation (31).In this example, the concentration of GeO₂ is assumed to be 7.9 molepercent at the center of the core and the concentration of B₂ O₃ is 13.5mole percent at the cladding. Therefore, control circuits 37 and 38 areprogrammed to produce dopant concentration profiles in the preform whichwill yield concentration profiles in the waveguide given by Equation(31).

Analog circuits which produce such control signals are well known. Forexample, "ANALOG COMPUTATION IN ENGINEERING DESIGN," Rogers andConnolly, McGraw-Hill Book Company, Inc., 1960, describes such circuits.However, in the preferred embodiment of the invention, a digitalmicroprocessor is used to generate the control signals. One example of amicroprocessor which is suitable for this purpose is the Program LogicController, manufactured by Allen-Bradley Company.

For the optical waveguide made in this manner, the pulse dispersionversus wavelength has been calculated and is shown by C2 in FIG. (4). Bycomparing C2 of the present invention to C1 for a single α profile, itcan be seen that curve C2 minimizes pulse dispersion over asignificantly broader range of wavelengths.

In FIG. 5, α₁ (λ) and α₂ (λ) are the functions of Eq. (2) which havebeen determined in Example 2. It can be seen that both α₁ and α₂ havezero slope at 0.85 microns in accordance with the condition of equation(14). It is because of the zero slope in α₁ (λ) and α₂ (λ) at 0.85microns that the pulse dispersion versus wavelength has a very broadregion of minimum pulse dispersion.

In FIG. (6), the GeO₂ and B₂ O₃ dopant profiles of this example areshpwn by curves G2 and B2. For comparison, the GeO₂ and B₂ O₃ dopantprofiles required to produce the optimal single α profile of the priorart are shown by curves G1 and B1. The prior art concentration profilesshown by G1 and B1 are given as

    C.sub.G (r)=0.079[1-(r/a).sup.1.957 ]                      (32)

    C.sub.B (r)=0.135(r/a).sup.1.957.                          (33)

It is clear that the dopant profile of the optimal single α profile andthe double α profile of this example are different. These differencescan be observed by using an X-ray microprobe technique to measure dopantprofiles of optical waveguides or of optical waveguides preforms. Such amicroprobe technique has been described by W. T. Kane in an articleentitled, "APPLICATIONS OF THE ELECTRON MICROPROBE IN CERAMICS AND GLASSTECHNOLOGY" which appears in Microprobe Analysis edited by C. A.Andersen, John Wiley & Sons, Inc. 1973.

EXAMPLE 3

As another illustration of a preferred embodiment of the invention,index profiles will be determined which provide minimum pulse dispersionat two wavelengths, λ₁ =0.80 microns and λ₂ =1.20 microns.

Let the symbols defined in equations (16)-(22) represent quantities at0.80 microns. Define an analogous set of quantities evaluated at 1.20microns and denote these quantities by writing a bar over the symbol.For example

    δ.sub.G =n.sub.c.sup.2 -n.sub.s.sup.2                (34)

where n_(C) and n_(S) are evaluated at 1.20 microns. The condition to besatisfied is that

    α.sub.1 =α.sub.1

    α.sub.2 =α.sub.2.                              (35)

By using equation (4) and the definitions of equations (16)-(22) itfollows that equation (35) is equivalent to the expression:

    (1-X.sub.i).sup.2 B.sub.ii +(1-X.sub.i)X.sub.j B.sub.ij +X.sub.j.sup.2 B.sub.jj =0                                               (36)

where i=G and j=B or j=G and i=B. In writing Eq. (36), small correctionterms of order X³ have been neglected. The quantities B_(ij) are definedas

    B.sub.ij =2(Z-Z)(δ.sub.i δ.sub.j +δ.sub.j δ.sub.i)+(2-Z)(δ.sub.i δ.sub.j '+δ.sub.j δ.sub.i ')-(2-Z)(δ.sub.j δ.sub.i '+δ.sub.i δ.sub.j ')                                          (37).

The coefficients B_(GG), B_(BB), B_(GB) can be evaluated from theaforementioned Sellmeier fits.

Equation (36) can then be solved to find

    X.sub.G =0.398

    X.sub.B =-2.021                                            (38)

and

    α.sub.1 =3.028

    α.sub.2 =1.581.                                      (39)

Pulse dispersion versus wavelength for a waveguide fabricated accordingto the above specifications is shown in FIG. 7. The pulse dispersion isminimized in the vicinity of 0.80 μm and 1.20 μm.

The dopant profiles of this waveguide are given as

    C.sub.G (R)=0.079[1-0.398(r/a).sup.3.028 -0.602(r/a).sup.1.581 ]

    C.sub.B (R)=0.135[3.021(r/a).sup.1.581 -2.021(r/a).sup.3.028 ]. (40)

If the linear approximation of equation (21) is not valid, the analysisbecomes more difficult; but the principles of waveguide design remainthe same.

Further improvement in reducing pulse dispersion can be achieved byintroducing additional design parameters ε_(i) into equation (13) sothat ##EQU24## The ε_(i) are considered to be small parameters, that is

    |ε.sub.i |≦2. i=l . . . N. (42)

Using standard methods of analysis ε_(i) can be chosen which furtherreduce dispersion. The exact values of ε_(i) depend on Δ_(j), (dΔ_(j)'/dλ), λ, n_(c), N_(c) and the distribution of optical power among themode of the waveguide.

Any particular choice of ε_(i) subject to the condition of equation (42) is within the scope of this invention.

While a particular embodiment of the invention has been shown anddescribed, various modifications are within the true spirit and scope ofthe invention. The appended claims are intended to cover all suchmodifications.

What is claimed is:
 1. An optical waveguide comprising at least threeglass-forming compounds and having a core with a radially-gradedrefractive index profile and a cladding, said refractive index profilechanging as a function of radius r substantially as: ##EQU25## where N≧2is the number of α-type index profile termsn_(c) is the refractive indexat r=o n_(o) is the refractive index at r=a Δ=(n_(c) ² -n_(o) ²)/2n_(c)² ##EQU26## and Δ_(i) and α_(i) are values which produce reduced pulsedispersion.
 2. The optical waveguide recited in claim 1 wherein thevalues α_(i) are given by: ##EQU27## where λ is a wavelength at whichsaid waveguide will be used and ##EQU28##
 3. The optical waveguiderecited in claim 1 wherein the values of α_(i) are given by: ##EQU29##where λ is a wavelength at which said waveguide will be used, ##EQU30##and where the ε_(i) produce improved pulse dispersion.
 4. The opticalwaveguide recited in claim 3 wherein Δ_(i) are values which producereduced pulse dispersion over a range of wavelengths.
 5. The opticalwaveguide recited in claim 3 wherein Δ_(i) are values which producereduced pulse dispersion at two or more wavelengths.
 6. The opticalwaveguide recited in claim 4 wherein Δ_(i) are values such that:##EQU31## where λ_(o) is a wavelength in the spectral range at whichsaid waveguide will be used.
 7. The optical waveguide recited in claim 6wherein ε_(i) =1 for i=1 . . . N.
 8. The optical waveguide recited inclaim 3 wherein the values of Δ_(i) are such that

    α.sub.i (λ.sub.1)=α.sub.i (λ.sub.2) . . . =α.sub.i (λ.sub.q),   i=1 . . . N where 2≦q≦N

where λ₁, λ₂ . . . λ_(q) include at least one wavelength at which saidwaveguide will be used.
 9. The optical waveguide recited in claim 9wherein ε_(i) =1 for i=1 . . . N.
 10. An optical waveguide comprising pglass forming compounds where p≦3 and having a core with aradially-graded composition profile and a cladding, said concentrationprofile C_(j) (r) of the glass forming compounds varying substantiallyas: ##EQU32## where the coefficients C_(ji) and α_(i) produce reducedpulse dispersion, and where C_(j) (o) denote the concentrations at theradius r=o .
 11. In an optical communication system comprising:a sourceproducing pulses of light having a mean wavelength λ, a multimodeoptical waveguide having input and output ends, said input end beingdisposed in light receiving relationship with respect to said source,and means responsive to light radiating from the output end of saidwaveguide, said optical waveguide comprising:a transparent core having aradius including at least three glass-forming compounds having arefractive index n_(c) at the center of said core, a layer of claddingmaterial surrounding said core, the refractive index of said layer beingless than the refractive index of said core, the improvement wherein therefractive index n(r) of said core varies as a function of the radialdistance r from the center of said core substantially as: ##EQU33##where α_(i) are values which produce minimum pulse dispersion in saidwaveguide and which are related to the properties of said glass formingcompounds such that: ##EQU34## n_(c) is the refractive index at thecenter of said core, n₀ is the refractive index at r=a, N_(c) =n_(c)-λdn_(c) /dλ and ##EQU35##